Definition:General Logarithm/Common

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Definition

Logarithms base $10$ are often referred to as common logarithms.


Examples

Common Logarithm: $\log_{10} 2$

The common logarithm of $2$ is:

$\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

This sequence is A007524 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Common Logarithm: $\log_{10} 3$

The common logarithm of $3$ is:

$\log_{10} 3 = 0.47712 \, 12547 \, 19662 \, 43729 \, 50279 \ldots$

This sequence is A114490 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Common Logarithm: $\log_{10} e$

The common logarithm of Euler's number $e$ is:

$\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03251 \, 82765 \, 11289 \, 18916 \, 60508 \, 22943 \, 97005 \, 803 \ldots$

This sequence is A002285 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Common Logarithm: $\log_{10} \pi$

The common logarithm of $\pi$ is:

$\log_{10} \pi = 0.49714 \, 98726 \, 94133 \, 85435 \, 12683 \ldots$

This sequence is A053511 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also known as

Common logarithms are sometimes referred to as Briggsian logarithms or Briggs's logarithms, for Henry Briggs.


In elementary textbooks and on most pocket calculators, $\log$ is assumed to mean $\log_{10}$.

This ambiguous notation is not recommended, particularly since $\log$ often means base $e$ in more advanced textbooks.


Also see

  • Results about logarithms can be found here.


Historical Note

Common logarithms were developed by Henry Briggs, as a direct offshoot of the work of John Napier.


After seeing the tables that Napier published, Briggs consulted Napier, and suggested defining them differently, using base $10$.

In $1617$, Briggs published a set of tables of logarithms of the first $1000$ positive integers.

In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places.


Before the advent of cheap means of electronic calculation, common logarithms were widely used as a technique for performing multiplication.


Sources